Content:
Part 1
1.1Short review of the classical theory of finite-dimensional integrable systems
1.2 Be-Hamiltonian Geometry and Lax pair
1.3 The Toda system
1.4 The Korteweg de Vries equation: direct and inverse scattering on the line with decreasing initial data
1.5 Long time asymptotic for the solution of the KdV equation with decreasing initial data and Deift-Zhou steepest descent method
1.6 The Cauchy problem for the KdV equation with periodic initial data and action-angle variables.
Part 2 Riemann Surfaces
2.1 Definition, examples, and topological properties
2.2 Holomorphic and meromorphic functions and differentials on a Riemann surface.
2.3 Jacobi variety and Abel theorem
2.4 Riemann Roch theorem and applications
2.5 Theta function and Riemann vanishing theorem
2.6 Baker-Akhiezer function and periodic solution of integrable nonlinear PDEs
